What is the range and domain, x and y intercepts, symmetry and asymptote of this given equation, f(x) = 2x^2 - 8 / x-1
all polynomials have a domain of (-∞,∞)
Rational functions do too, except where the denominator is zero. So,
f(x) has a domain of (-∞,1)U(1,∞)
Near x=1, 2x^2-8 is near -6.
So, for
x<1 y->∞
x>1 y->-∞
So, f(x) has a range of (-∞,∞)
2x^2-8 = 2(x-2)(x+2)
Since a fraction is zero when the numerator is zero, y=0 at x=±2
f(0) = 8
As or symmetry, this is a rotated hyperbola, rotated through an angle of -13.28°
To determine the range and domain of the given equation f(x) = (2x^2 - 8) / (x - 1), we need to consider the restrictions and limitations of the function.
1. Domain: The domain represents all possible x-values for which the function is defined. In this case, the function is defined for all values of x except those that would result in a division by zero. Therefore, x cannot be equal to 1. Thus, the domain is all real numbers except x = 1.
2. Range: The range represents all possible y-values that the function can take. As x approaches positive or negative infinity, f(x) also approaches infinity. Hence, the range of the function is all real numbers except zero.
To find the x-intercept(s) of the equation (where the graph intersects the x-axis), we set f(x) equal to zero and solve for x. In this case:
0 = (2x^2 - 8) / (x - 1)
To solve this equation, we can multiply both sides by (x - 1) to eliminate the denominator:
0(x - 1) = 2x^2 - 8
0 = 2x^2 - 8
We can now solve for x using factoring, completing the square or the quadratic formula. Factoring is the simplest approach here:
0 = 2(x^2 - 4)
Setting each factor equal to zero, we have:
x^2 - 4 = 0
This quadratic equation factors as:
(x + 2)(x - 2) = 0
Setting each factor equal to zero gives us the x-intercepts:
x + 2 = 0 --> x = -2
x - 2 = 0 --> x = 2
So, the x-intercepts of the given equation are -2 and 2.
To find the y-intercept (where the graph intersects the y-axis), we set x equal to zero and solve for f(x):
f(0) = (2(0)^2 - 8) / (0 - 1)
f(0) = -8 / -1
f(0) = 8
Therefore, the y-intercept is 8.
To determine the symmetry of the graph, we need to check if the equation is even or odd. If the equation is even (f(x) = f(-x)), the graph is symmetric with respect to the y-axis. If the equation is odd (f(-x) = -f(x)), the graph is symmetric with respect to the origin.
In this case, we substitute -x into the function and check if it is equal to f(x):
f(-x) = [2(-x)^2 - 8] / (-x - 1)
= (2x^2 - 8) / (-x - 1)
= -((2x^2 - 8) / (x + 1))
Therefore, f(-x) is not equal to -f(x), and f(x) is not equal to f(-x). Hence, the graph does not possess symmetry with respect to the origin or the y-axis.
Lastly, let's determine the asymptote(s) of the function. Asymptotes are lines that the graph of the function approaches but never touches. In this case, there are two types of asymptotes: horizontal and vertical.
Vertical asymptotes occur when the denominator of a rational function equals zero. In this equation, the denominator is x - 1. Therefore, the vertical asymptote is x = 1.
Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the numerator has a degree of 2 while the denominator has a degree of 1. Therefore, there is no horizontal asymptote.
To summarize:
- Domain: All real numbers except x = 1.
- Range: All real numbers except zero.
- x-intercepts: -2 and 2.
- y-intercept: 8.
- Symmetry: No symmetry with respect to the y-axis or the origin.
- Vertical asymptote: x = 1.
- Horizontal asymptote: None.